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A Galois-Theoretic Complexity Measure for Solving Systems of Algebraic Equations

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arxiv 2503.17884 v1 pith:GENJU7CJ submitted 2025-03-22 math.AG cs.CC

classification math.AGcs.CC
keywords algebraiccomplexityequationsgaloisgeometryquantitysolvingwidth
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Motivated by applications of algebraic geometry, we introduce the Galois width, a quantity characterizing the complexity of solving algebraic equations in a restricted model of computation allowing only field arithmetic and adjoining polynomial roots. We explain why practical heuristics such as monodromy give (at least) lower bounds on this quantity, and discuss problems in geometry, optimization, statistics, and computer vision for which knowledge of the Galois width either leads to improvements over standard solution techniques or rules out this possibility entirely.

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  1. Numerically Computing Galois Groups of Minimal Problems

    cs.CV 2025-07 conditional novelty 4.0 of 10

    A tutorial arguing that the Galois group and the Galois-width invariant characterize the intrinsic algebraic difficulty of minimal problems, with numerical monodromy code for the five-point problem.

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